Rigidity of Lie foliations with locally symmetric leaves

TL;DR

This paper proves that minimal Lie foliations with leaves locally isometric to certain symmetric spaces are smoothly conjugate to homogeneous foliations, extending Zimmer's theorem and establishing new rigidity results for Riemannian foliations.

Contribution

It generalizes Zimmer's theorem by showing such foliations are conjugate to homogeneous ones, and extends arithmeticity and rigidity results for foliations with symmetric leaves.

Findings

Foliations with symmetric leaves are smoothly conjugate to homogeneous foliations.

Extension of Zimmer's arithmeticity theorem for holonomy groups.

Rigidity theorem for Riemannian foliations with locally symmetric leaves.

Abstract

We prove that if the leaves of a minimal Lie foliation are locally isometric to a symmetric space of non-compact type without a Poincare disk factor, then the foliation is smoothly conjugate to a homogeneous Lie foliation up to finite covering. This result generalizes and strengthens Zimmer's theorem, which characterizes minimal Lie foliations with leaves isometric to a symmetric space of non-compact type without real rank one factors as pullbacks of homogeneous foliations. As applications, we extend Zimmer's arithmeticity theorem for holonomy groups and establish a rigidity theorem for Riemannian foliations with locally symmetric leaves.